Embedding theorems for non-positively curved polygons of finite groups
type: Item
instance of: scholarly article
title: Embedding theorems for non-positively curved polygons of finite groups
zbMATH Open document ID: 0892.20026
DOI: 10.1016/S0022-4049(96)00077-1
author: Tim Hsu
author: Daniel T. Wise
published in: Journal of Pure and Applied Algebra
publication date: 1998-06-09T00:00:00Z
review text: A polygon of groups has groups associated to its vertices, its edges, and to its face. There is an inclusion of the face group into each edge group and an inclusion of each edge group into its associated vertex groups. It is assumed that at each vertex the two edge groups intersect only in the face group. The fundamental group of the polygon of groups is defined to be the free product of the vertex groups amalgamated along the edge groups. A concept of angle between the two edge groups at a vertex has been defined by Gersten and Stallings. If the resulting angles belong to a Euclidean or hyperbolic polygon, then the polygon of groups is called nonpositively curved. Gersten and Stallings proved that for a nonpositively curved polygon of groups, the vertex groups embed in the amalgamated free product. K. S. Brown proved that Thompson's infinite simple group is the fundamental group of a polygon of groups, and hence such groups need not be residually finite. That example, however, has positive curvature. In the paper under review, the authors show that there is a nonpositively curved triangle of finite groups whose fundamental group is not residually finite. To construct their examples, they prove two imbedding theorems that are of independent interest. The first states that if \(X\) is a complete squared complex, i. e. a 2-complex whose universal cover is isomorphic to the product of two trees, then for each prime \(p\) there is a right-angled square of finite \(p\)-groups \(R_p(X)\) such that \(\pi_1(X)\) imbeds into \(\pi_1(R_p(X))\). The second states that if \(R\) is a square of finite groups with all vertex angles \(\leq{\pi\over 2}\), then \(\pi_1(R)\) can be imbedded in the fundamental group of a nonpositively curved triangle of finite groups. Since there exist complete squared complexes with nonresidually finite fundamental groups, applying the two imbedding theorems yields the desired examples.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1121667
zbMATH Keywords: fundamental groups
zbMATH Keywords: polygons of groups
zbMATH Keywords: squared complexes
zbMATH Keywords: residually finite groups
zbMATH Keywords: triangles of groups
zbMATH Keywords: squares of groups
zbMATH Keywords: angles between subgroups
zbMATH Keywords: embedding theorems
zbMATH Keywords: amalgamated free products
reviewed by: Darryl McCullough
MaRDI profile type: Publication
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full work available at URL: s0022-4049(96)00077-1
OpenAlex ID: W1975766557
description: scientific article; zbMATH DE number 1121667